Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 245 
X, y selected from 0, 1, . . . , p — 1, provided n is a quadratic non-residue of the 
prime p. Then x-\-y\/n is a root of p+^=l (mod p), which therefore has 
p + 1 complex roots, all a power of one root. There is a table of indices for 
these roots when p = 29 and p = 41. [Lebesgue.^^] 
A. E. Pellet^^ considered the product A of the squares of the differences 
of the roots of a congruence /(x) = (mod p) having no equal roots. Then 
A is a quadratic non-residue of p if f{x) has an odd number of irreducible 
factors of even degree, a quadratic residue if f{x) has no irreducible factor 
of even degree or has an even number of them. For, if 5i, . . ., 5^ are the 
values of A for the various irreducible factors of f{x), then A=a^di. . .5j 
(mod p), where a is an integer. Hence it suffices to consider an irreducible 
congruence /(a:) = (mod p). Let v be its degree and i a root. In 
v-i i-i 
!/=n n {x'' -x^) 
1=1 A; = 
replace x by the v roots; we get two distinct values if v is even, one if ;^ is 
odd. In the respective cases, i/^=A (mod p) is irreducible or reducible. 
R. Dedekind^^ noted that, if P(x) is a prime function of degree / modulo 
p, a prime, a congruence F{x) = (mod p, P) is equivalent to the congruence 
F(a) = (mod tt), where tt is a prime ideal factor of p of norm p^, and a is 
a root of P(a) = (mod tt). 
A. E. Pellet^^ denoted by/(a;)=0 the equation of degree ^(A;) having 
as its roots the primitive A;th roots of unity, and by /i(i/) =0 the equation 
derived by setting y = x-\-\/x. If p is a prime not dividing k, f{x) is con- 
gruent modulo p to a product of <f>{k)/v irreducible factors whose degree v 
is the least integer for which p" — ! is divisible by k. li fi{y) = (mod p) 
has an integral root a,f(x) is divisible modulo p by x^ — 2ax-\-l. Either the 
latter has two real roots and f{x) and fi{y) have all their roots real and 
p — 1 is divisible by k, or it is irreducible and f(x) is a product of quadratic 
factors modulo p and the roots oifi{y) are all real and p+1 is divisible by k. 
If k divides neither p+l nor p — l,fi{y) is a product of factors of equal 
degree modulo p. [Cf. Sylvester,^^ etc., Ch. XVI.] 
Let A; be a divisor f^ 2 of p + L Let X be an odd number divisible by no 
prime not a factor of k, and relatively prime to{p+l)/k. Then x^^ — 2ax^ + 1 
is irreducible modulo p [Serret,^^ No. 355]. Also, if h is not divisible by p 
F={x-\-by''-2aix^-b^)^+ix-by^ 
is irreducible modulo p; replacing x^ by y, we obtain a function of degree X 
irreducible modulo p. If /c is a divisor ?^2 of p — 1 and if X is odd, prime to 
{p — l)/k and divisible by no prime not a factor of k, F decomposes modulo 
p into two irreducible functions of degree X. 
The function /(x^) is either irreducible or the product of two irreducible 
factors of degree v. In the respective cases, the product A of the squares of 
"Comptes Rendus Paris, 86, 1878, 1071-2. 
"Abhand. K. Gesell. Wiss. Gottingen, 23, 1878, p. 25. Dirichlet-Dedekind, Zahlentheorie, ed. 
4, 1894, 571-2. 
s^Comptes Rendus Paris, 90, 1880, 1339-41. 
